How to calculate the integral of a erf times another function?

[jollage]

Hi

I'm encountered the calculation of this function $\int^{\infty}_0 e^{-x^2} f(x) dx$. How to do it? Thanks.

 

[SteamKing]

Depends of what f(x) is.

 

[jollage]

Depends of what f(x) is.

SteamKing, thanks. I have another problem, which I encountered in the same paper. Maybe you could help me. Thanks in advance.

The author asserts that $V(k,t)=V(k,0)e^{-k^2t}+ k\int^{t}_{0}C(t')e^{k^2(t'-t)}dt'$ implies $V(k,t)=C(t)/k + \mathcal{O}(k^{-3})$ when t>0 and $k\rightarrow \infty$. Could you see this?

 

[mathman]

I haven't worked out the details.
It looks like the V(k,0) term -> 0 very fast, so it can be ignored.
The lntegrand of the integral looks as if -> δ(t'-t)/k².

 

[jollage]

I haven't worked out the details.
It looks like the V(k,0) term -> 0 very fast, so it can be ignored.
The lntegrand of the integral looks as if -> δ(t'-t)/k².

Mathman, thanks. I agree with you, but I need the detail to understand it.. Since it seems it is impossible to expand the exponential term in the integral, I don't know other techniques to evaluate the integral...